energy_pos
plain-language theorem explainer
The theorem establishes that the energy per resolution equals phi to the fifth power and is strictly positive. Cosmologists deriving dark energy equations of state and vacuum uniformity certificates cite this result to guarantee positive energy densities. The proof is a one-line term application of the positivity lemma for powers of the golden ratio.
Claim. $0 < phi^5$, where the energy per resolution is defined as this quantity and $phi$ is the golden ratio.
background
In the Recognition Science module on the Bremermann limit, the energy per resolution is the minimum energy quantum for one recognition event. It equals phi^5 because the reduced Planck constant is set to phi^{-5} in native units, and the 8-tick cycle sets the fundamental resolution period. The upstream theorem from OscillatoryDynamicsDeep proves energy levels are positive by factoring into terms involving omega_pos and phi^{-5}.
proof idea
The proof is a one-line term that applies the lemma pow_pos to the positivity of phi and the exponent 5.
why it matters
This result supplies the positivity hypothesis required by ConstantEnergyContribution and w_eq_neg_one to obtain the dark energy equation of state w = -1. It also feeds vacuumUniformityCert and lattice-state constructions in ethical applications. The declaration closes the positivity step for the phi^5 quantum tied to the eight-tick octave in the Bremermann bound.
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